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Not any time soon. By far the greatest challenge is getting the DNA sequence from the computer into a form that could be implanted into an embryo. Craig Venter's team was able to build a bacterial genome from scratch, but that was incredibly painstaking work, requiring the assembly of many small fragments of DNA into a single circular chromosome. With current technology that process becomes prohibitively difficult for a human-like genome. You would have to build each chromosome from scratch, these chromosomes are linear, and you'd need to associate with them the various histones and other proteins help tightly fold them together.

Another huge problem is epigenetics. The next generation sequencing methods used for the neanderthal genome just give you DNA sequences, not what epigenetic modifications are present on that DNA. Genomic imprinting in mammals is very important, and presents problems even in normal cloning efforts where genomes are taken wholesale from live animals.

Not to mention that 1% of the current genome is contaminated with human DNA (not meant as a criticism, its amazing what they have been able to do with a toe bone if this work stands up to peer review).

And none of this takes into account the ethical problems associated with this. Would you clone it in a human? Chimp? Either way its got serious ethical questions.

Dont take my word for it though, here is the leader of the project commenting on this question a few years ago:

Paabo said because the Neanderthal DNA was scattered in imperfect
fossils, the notion of cloning a Neanderthal was far-fetched.

"Starting from the DNA extracted from a fossil, it is and will remain
impossible," he said. "There is not really an improvement on current
technologies that would make that possible.